If Mozilla had renamed Phoenix to Postgres, then they would have a problem. But they didn't. They chose Firebird, which is a fairly common term used for a variety of products. Just as the team who created the Firebird database must have realized when choosing the name the risk of a future more widely known software product coming along bearing the same name in a different domain, the Mozilla team must also realize the risk of the same thing happening to them in the future.

Both Firebird database and Mozilla Firebird teams probably picked "Firebird" because they liked the name. If either one of them had wanted an original name, they should have choosen something more original such as "Postgres" or "Mozilla."